Cellular Automata (CA) provide an interesting computational model for designing several cryptographic primitives, especially in the symmetric-key setting. Indeed, during the past decades CA have been employed to define pseudorandom generators for stream ciphers, as in the seminal work by Wolfram in the 80s, or S-boxes for block ciphers, such as the rule χ used by Daemen et al. in the design of the Keccak sponge construction. In this talk, we will survey recent results on combinatorial problems related to cryptographic primitives based on CA, focusing on the examples of orthogonal arrays, bent functions and S-boxes. We will also present counting and enumeration problems that are still open for each of these examples, proposing some approaches to address them in future research.